Supplementary Information for Quantitative analysis of spin wave dynamics in ferrimagnets across compensation points Eloi Haltz, João Sampaio, Sachin Krishnia, Léo Berges, Raphaël Weil, Alexandra Mougin, and André Thiaville I. REORIENTATION RANGE IN FERRIMAGNETS spinflop J = 80 MJ/m³ 1 15 20 II. 25 30 35 40 45 50 BRILLOUIN LIGHT SCATTERING To measure the SW dynamics, we performed BLS spectrometry (see Fig. 1(b)), whereby a laser beam is backscattered by the sample and its spectrum is analyzed. The sample, installed on a temperature-controlled holder, is illuminated by a laser beam linearly polarized in ŷ (s-polarized) and the backscattered light polarized in the (x̂, ẑ) plane (ppolarized) is analyzed with a tandem Fabry-Perot spectrometer (The Table Stable Ltd, model TFP-2 HC). Through the Stokes (S) and the anti-Stokes (AS) processes, the incident photons can generate and absorb SW modes with a wave vector kS k −x̂ or kAS k x̂, respectively. The magnitude of kS/AS is fixed by the angle of incidence θB = 30° and the light wavelength λ = 532 nm such as −kS = kAS = (4π sin(θB )/λ )x̂ = (11.8 µm−1 )x̂. The frequency of the backscattered light is shifted by the SW frequency, producing peaks at fAS > 0 and fS < 0 in the BLS spectrum. A magnetic field H along ŷ (Fig. 1(b)) is applied to counteract the perpendicular anisotropy and measure the SW in the Damon-Eshbach geometry (k k x̂ and m k ŷ) . This geometry gives a larger signal, allows easily applying a magnetic field, and is convenient to measure the SW chirality [17,18]. The points in the center of the spectra ( f ≈ 0) correspond to the reference unscattered light and were not considered in the analysis. All shown spectra were normalized to their maximum point. In some spectra at the lowest temperatures, the HF peaks do not appear due to the restricted measurement range of 55 ±30 GHz instead of ±100 GHz. This was chosen to reduce µ₀H [T] mz,TM 0 H H > HK mRE mTM H < HK 40 MJ/m³ µ₀H [T] 10 In ferrimagnets with out-of-plane anisotropy, the range of in-plane field H for which the magnetization is aligned with the field varies with temperature T . The Fig. S1 shows the reorientation region for different values of the inter-lattice exchange (and for the same other parameters as in the main text). In the limit of small H in comparison with the interlattice exchange, the sublattice moments remain antiparallel and the magnetization is out-of-plane for H < HK (T ), where HK (T ) can be estimated with the formula for ferromagnets (HK (T ) = µ M2K(T ) − MS (T )). Near TMC and for larger H, 0 S the field induces a tilt between the two sublattices, creating a transverse net magnetic moment that aligns with H. This pushes the two moments to be perpendicular to the field, in a similar way to the spin-flop transition in antiferromagnets. The border of the gray region can be obtained analytically, although its formula is cumbersome. In our system, these two behaviors coexist, as is shown by the gray region in Fig. 3(e). TMC 20 MJ/m³ µ₀H [T] 5 T [K] Figure S1. Diagram of the reorientation range versus in-plane field H and temperature T in a RE-TM ferrimagnetic alloy for inter-lattice exchange J values above, at, and below the fitted value. The gray background corresponds to the out-of-plane component of the TM magnetization. The dotted line shows HK (T ) using the effective ferromagnet formula (equal in the three plots). The dashed line shows the spin-flop boundaries HJ (T ) calculated by neglecting the anisotropy (different in the three plots). Vertical lines correspond to TMC and TAC . the instrumental broadening when measuring the sharper LF peaks (instrumental full-width/half-maximum of ∼ 1.2 GHz instead of ∼ 4 GHz). III. ADDITIONAL DATA Fig. S2 shows AHE hysteresis cycles measured in a quasiin-plane geometry at different temperatures, used to determine 2 5 Figure S2. Anomalous Hall effect loops under field in quasi-in-plane geometry (83° to the normal of the film) for different temperatures. HK . The AHE signals were measured in a separate sample of the film, which was lithographically patterned into Hall crosses. After the patterning process, the TMC of the AHE sample decreased by 30 K. This decrease was corrected in the temperature values of the HK data. Figs. S3 and S4 show additional measurements of the mode frequencies versus temperature or field, similar to main text but at different constant field or temperature, respectively. 3 Figure S3. Top panel: BLS peak centers and widths versus T for different applied fields H. Similarly to Fig. 2(a), peak centers f0 (points) and widths δ (vertical bars) are extracted from fits. Lines and shaded regions are the frequency and peak width calculated with the Smit-Beljers method. Bottom panel: Frequency difference ∆ f = | fAS | − | fS | for the LF and HF peaks versus T similarly to Fig. 2(d). The lines are the prediction of the Smit-Beljers method. The data for H=0.56 T are the same shown Fig. 2. Figure S4. Top panel: BLS peak centers and widths versus H for different temperatures. Similarly to Fig 3(a), peak centers f0 (points) and widths δ (vertical bars) are extracted from fits. Lines and shaded regions are the frequency and peak width calculated with the Smit-Beljers method. Bottom panel: Frequency difference ∆ f = | fAS | − | fS | for the LF and HF peaks versus H similarly to Fig 3(b). The lines are the prediction of the Smit-Beljers method. The data for T =367 K are the same shown in Fig 3(a,b).
